A great breakthrough in studying the evolution and demographics of the cosmic X-ray background (CXB) population is the Nuclear Spectroscopic Telescope Array (NuSTAR), the first focusing X-ray observatory with high sensitivity at >10keV and thus at the CXB peak (~20-30keV). Here we present results from (1) the NuSTAR serendipitous survey, the largest area (~7 deg^2) component of the NuSTAR extragalactic survey programme, and (2) pointed NuSTAR observations of heavily obscured, candidate Compton-thick Type 2 quasars (CTQSO2s). (1): The serendipitous survey source statistics are good for a high-energy X-ray survey. The population sampled is broadly similar to the nearby high-energy selected AGNs sampled by Swift/BAT, but scaled up in redshift, luminosity and mass. (2): We find that for candidate CTQSO2s at z < 0.5 NuSTAR provides an improvement compared to previous constraints in the literature which are limited to low energies (<10keV; Chandra and XMM-Newton), as higher column densities (by factors of ~2.5-1600) and higher luminosities (by factors of ~10-70) can now be reliably constrained. We extrapolate these results to the parent population in order to make inferences about the column density (NH) distribution of optically-selected QSO2s.